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#### Mathelogician

##### Member

- Aug 6, 2012

- 35

We have a theorem in natural deduction as follows:

Let H be a set of hypotheses:

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**H U {~phi) is inconsistent => H implies (phi).**

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Now the question arises:

Let H={p0} for an atom p0. So H U{~p0}={p0 , ~p0}.

We know that {p0 , ~p0} is inconsistent, so by our theorem we would have:

{p0} implies ~p0.

Which we know is impossible.(because for example it means that ~p0 is a semantical consequence of p0).

Now what's wrong here?

Thanks